Here the main contribution does not come from a singularity, but rather from a small neighborhood of a saddle point (if \(\log H/\log y\)  grows sufficiently fast with respect to \(y\) ) or is simply not localized (if \(\log H/\log y\)  is small).
This is in analogy with studying \(y\) -friable numbers up to \(H\)  via Perron's formula, see [1]} and [2]}.
